{"id":50,"date":"2017-08-23T07:12:20","date_gmt":"2017-08-23T07:12:20","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=50"},"modified":"2017-09-18T15:05:32","modified_gmt":"2017-09-18T15:05:32","slug":"graphing-linear-equations-and-inequalities","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/graphing-linear-equations-and-inequalities\/","title":{"rendered":"Graphing Linear Equations and Inequalities"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-linear-equations-inequalities\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/finding-the-equation-of-a-line\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Graphing Linear Equations and Inequalities<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will study how to graph a linear equation, a linear inequality, or a system of inequalities.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>In a <em>linear\u00a0equation<\/em>, no variable has an exponent greater than\u00a0one.<\/li>\n<li>In <em>standard form<\/em>,\u00a0a linear equation in two variables is in the form <em>ax + by = c<\/em>, where\u00a0<em>a, b, <\/em><span style=\"text-decoration: none;\">and\u00a0<\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">are constants, and not all are equal to zero. <\/span><\/li>\n<li>In a <em><strong>linear inequality<\/strong><\/em>, the\u00a0symbol\u00a0<em>&gt; <\/em> means <em><span style=\"text-decoration: none;\">greater than<\/span><\/em><span style=\"text-decoration: none;\">, \u00a0<\/span><em><span style=\"text-decoration: none;\">&lt; <\/span><\/em><span style=\"text-decoration: none;\"> means <\/span><em><span style=\"text-decoration: none;\">less than<\/span><\/em><span style=\"text-decoration: none;\">, \u00a0<\/span><em><span style=\"text-decoration: none;\">\u2264 <\/span><\/em><span style=\"text-decoration: none;\"> means <\/span><em><span style=\"text-decoration: none;\">less than or equal to<\/span><\/em><span style=\"text-decoration: none;\">, and <\/span><em><span style=\"text-decoration: none;\">\u2265 <\/span><\/em><span style=\"text-decoration: none;\"> means <\/span><em><span style=\"text-decoration: none;\"> greater than or equal to<\/span><\/em><span style=\"text-decoration: none;\">. <\/span><\/li>\n<\/ul>\n<section>\n<h3><strong>How do we graph a linear equation?<\/strong><\/h3>\n<p>One method for graphing a <abbr title=\"an algebraic expression of the form y = ax + b, where a and b are constants. When graphed on the x- and y- axes, this function produces a straight line with slope a and y-intercept b.\">linear\u00a0equation<\/abbr> is to solve for several ordered pairs, plot\u00a0the resulting points, and then connect the points.<\/p>\n<p>Another method is to find the two <abbr title=\"the point where a line crosses its respective axis\">intercepts<\/abbr>,\u00a0<em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0<\/span>An intercept is the point where a line crosses the respective\u00a0axis.<\/p>\n<p>For example, to graph the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image002.gif\" width=\"70\" height=\"14\" name=\"graphics3\" align=\"ABSBOTTOM\" border=\"0\" \/>,\u00a0first find the <em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-intercept.\u00a0Since this is the point where the line crosses the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0<\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">will equal 0. In the equation, set <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal to 0 and solve for <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image004.gif\" width=\"80\" height=\"59\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The <em><span style=\"text-decoration: none;\">x-<\/span><\/em>intercept\u00a0is the ordered pair (4, 0).<\/p>\n<p>Because the <em><span style=\"text-decoration: none;\">y-<\/span><\/em>intercept\u00a0is the point where the line crosses the\u00a0<em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0set <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal to zero and solve for <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s8_p2_html_499057c2.gif\" width=\"92\" height=\"73\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The <em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">in<\/span>tercept\u00a0is the point represented by the ordered pair (0, 2). Plot the\u00a0points and connect them to find the line.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20Art%20001.JPG\" width=\"305\" height=\"315\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>To find out if any other points are on the line, substitute the\u00a0values for <em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in<\/span>to the equation of the line and see whether the equation\u00a0is true.<\/p>\n<p>Recall that:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image008.gif\" width=\"174\" height=\"38\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>A linear equation can also be written in the form <em><span style=\"text-decoration: none;\">y\u00a0= mx + b, <\/span><\/em><span style=\"text-decoration: none;\">where <\/span><em><span style=\"text-decoration: none;\">m\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em>are constants.\u00a0This is called <abbr title=\" y = mx +b \">slope-intercept\u00a0form<\/abbr><span style=\"text-decoration: none;\">. The\u00a0symbol <\/span><em><span style=\"text-decoration: none;\">m\u00a0<\/span><\/em><span style=\"text-decoration: none;\">stands for the slope of the line, and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept.<\/span><\/p>\n<p>To put the previous equation into slope-intercept form, solve\u00a0for <em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image010.gif\" width=\"106\" height=\"125\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Because the slope of the line is negative, the line slopes\u00a0upward and to the left, indicating an inverse relationship between\u00a0the two variables. The slope is\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image012.gif\" width=\"22\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0meaning that every time <em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">increases by 1, <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">decreas<\/span>es by 2.<\/p>\n<p>To graph, first plot the <em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in<\/span>tercept, which is at the point (0, 2).<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20art%20002.JPG\" width=\"303\" height=\"315\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/><\/center>Then move 1 unit up and 2 units left from the <em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">intercept.\u00a0That gives us a second point, which we then connect to the\u00a0<\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0to get the line. <\/span><\/p>\n<p>For another example, graph the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image014.gif\" width=\"75\" height=\"14\" name=\"graphics11\" align=\"ABSBOTTOM\" border=\"0\" \/>\u00a0using the two intercepts.<\/p>\n<p>To do so, first find the <em><span style=\"text-decoration: none;\">x-<\/span><\/em><span style=\"text-decoration: none;\">intercept\u00a0by setting <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em>equal to 0.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image016.gif\" width=\"85\" height=\"83\" name=\"graphics12\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Then find the <em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0by setting <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal to 0. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image018.gif\" width=\"86\" height=\"61\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Finally, graph the two points and connect them.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20art%20003.JPG\" width=\"350\" height=\"315\" name=\"graphics14\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>For extra practice, graph the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image022.gif\" width=\"76\" height=\"14\" name=\"graphics15\" align=\"ABSBOTTOM\" border=\"0\" \/>\u00a0using the slope-intercept form.<\/p>\n<p>First, solve for <em><span style=\"text-decoration: none;\">y.<br \/>\n<\/span><\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image030.gif\" width=\"147\" height=\"125\" name=\"graphics16\" align=\"ABSBOTTOM\" border=\"0\" \/><\/p>\n<p>The <em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0is at the point (0, \u20133). Plot this point on the graph. The\u00a0slope is\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p2_clip_image026.gif\" width=\"9\" height=\"34\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so from the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0move 4 units right and 3 units up. Then connect the two points<\/span>.<\/p>\n<h4><strong>What happens if either of the coefficients in the\u00a0equation is zero?<\/strong><\/h4>\n<p>If the coefficient of <em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is zero, we have\u00a0an equation such as <\/span><em><span style=\"text-decoration: none;\">y\u00a0= <\/span><\/em><span style=\"text-decoration: none;\">5. This means that for all values of <\/span><em><span style=\"text-decoration: none;\">x,\u00a0y = <\/span><\/em><span style=\"text-decoration: none;\">5. Therefore,\u00a0the graph is a horizontal line at <\/span><em><span style=\"text-decoration: none;\">y\u00a0=<\/span><\/em><span style=\"text-decoration: none;\"> 5 . If the\u00a0coefficient of <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is zero, then you have a vertical line at the given value for <\/span><em><span style=\"text-decoration: none;\">x.<br \/>\n<\/span><\/em><\/p>\n<h4><strong>How do we graph an inequality?<\/strong><\/h4>\n<p>An inequality divides the <abbr title=\"a plane with a rectangular coordinate system that associates each point in the plane with a pair of numbers.\">Cartesian\u00a0plane<\/abbr> into two half-planes. Begin by graphing the\u00a0inequality as if it were an equation. If the inequality contains a\u00a0<em>\u2264\u00a0<\/em>or <em> \u2265\u00a0<\/em>symbol, the line will be graphed as a solid line. If the inequality contains a <em> &lt;\u00a0<\/em>or <em> &gt;\u00a0<\/em>symbol, the line will be graphed as a dashed line. After graphing\u00a0the line, shade in the half- plane that contains the values that\u00a0solve the inequality. To find these values, pick a point on one\u00a0side of the line (use the origin if possible). If the point makes\u00a0the inequality true, shade the side of the line containing the\u00a0point. If the point makes the inequality false, shade the opposite\u00a0side of the line, i.e., the side that does not include that point.<\/p>\n<p>For example, graph the inequality\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image002.gif\" width=\"62\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>First, graph the inequality as the equation <em><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image004.gif\" width=\"62\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0<\/em>To do so, find the <em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0by setting <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal to 0<\/span>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image006.gif\" width=\"62\" height=\"59\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The <em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-intercept\u00a0is (2, 0). <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Then find the <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">intercept\u00a0by setting <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal to 0. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image008.gif\" width=\"82\" height=\"61\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The <em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-i<\/span>ntercept\u00a0is (0, \u20138).<\/p>\n<p>Graph the two points and connect them with a line. Use a solid\u00a0line because the inequality contains the <em> \u2265 \u00a0<\/em>symbol.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20art%20005.JPG\" width=\"350\" height=\"434\" name=\"graphics12\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Finally, test to see which half-plane contains the values that\u00a0solve the inequality. Choose a point to one side of the line and\u00a0test the values. Use the point (4, 0).<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image012.gif\" width=\"71\" height=\"35\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The point (4, 0) lies to the right of the line and solves the\u00a0inequality, so we shade in the half-plane to the right of the\u00a0line.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20art%20006.JPG\" width=\"288\" height=\"412\" name=\"graphics14\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>For extra practice, graph the inequality\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image016.gif\" width=\"77\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Begin by graphing the inequality as an equation. Graph the line\u00a0by finding the intercepts.<\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image018.gif\" width=\"89\" height=\"205\" name=\"graphics16\" align=\"absmiddle\" border=\"0\" \/><\/em><\/p>\n<p>The two intercepts are (3, 0) and (0, 9). Graph the points and\u00a0connect with a dashed line. The line is dashed because the\u00a0inequality contains the <em> &lt;\u00a0<\/em>symbol.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20Art%20007.JPG\" width=\"345\" height=\"434\" name=\"graphics18\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Then test a point to see what half-plane to shade. Try the\u00a0point (0,0)<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p3_clip_image022.gif\" width=\"107\" height=\"39\" name=\"graphics19\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Because (0, 0) makes the equation true, shade in the half-plane\u00a0to the left of the line.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.2%20Art%20008.JPG\" width=\"345\" height=\"434\" name=\"graphics20\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<h4><strong>How do we graph two inequalities?<\/strong><\/h4>\n<p>We can solve a <abbr title=\"any set of simultaneous equations\">system<\/abbr> of two or more inequalities by following the steps above, but\u00a0repeated for each inequality.<\/p>\n<p>For example, graph the values that solve both <em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">&#8211;<\/span><em><span style=\"text-decoration: none;\"> y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">&gt; 3 and 5<\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">&#8211; 4<\/span><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">&lt;\u00a020. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">To do so, first find the <\/span><em><span style=\"text-decoration: none;\">x-\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y- <\/span><\/em><span style=\"text-decoration: none;\">intercepts\u00a0for each inequality, and graph the<\/span>m as equations.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p4_clip_image006.gif\" width=\"62\" height=\"182\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The intercepts of the first inequality are (3, 0) and (0, \u20133).<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p4_clip_image008.gif\" width=\"78\" height=\"157\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The intercepts of the second equation are (4, 0) and (0, \u20135).<\/p>\n<p>Graph both lines as dashed, since both contain the <em> &lt; \u00a0<\/em>symbol.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/math_mod_art009.jpg\" width=\"300\" height=\"300\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Because the original question asked us to shade in the areas\u00a0that solve both equations, we can deduce that any values that\u00a0solve both inequalities would lie between our two broken lines.\u00a0Test the conjecture by choosing a point between the two lines and\u00a0seeing if it solves both inequalities. In this case, choose point\u00a0(2, -2) and plug it into both lines.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p4_clip_image015.gif\" width=\"75\" height=\"62\" name=\"graphics7\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The point solves the first inequality.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s5_p4_clip_image017.gif\" width=\"100\" height=\"76\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The point also solves the second inequality. Therefore, shade\u00a0in the area between the two broken lines.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/math_mod_art0011.jpg\" width=\"300\" height=\"300\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-linear-equations-inequalities\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/finding-the-equation-of-a-line\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Graphing Linear Equations and Inequalities Objective In this lesson, we will study how to graph a linear equation, a linear inequality, or a system of inequalities. Previously Covered: In a linear\u00a0equation, no variable has an exponent greater than\u00a0one. In standard form,\u00a0a linear equation in two variables is in the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-50","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/50","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=50"}],"version-history":[{"count":13,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/50\/revisions"}],"predecessor-version":[{"id":729,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/50\/revisions\/729"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}